3.16 (a) what conditions are necessary before you can use a stream function to solve for the ow
eld?
(b) what conditions are necessary before you can use a potential function to solve for the ow eld?
(c) what conditions are necessary before you can apply Bernoulli’s equations to relate two points in a
ow eld?
(d) under what conditions does the circulation around a closed uid line remain constant with respect
to time?
a 2dimensional bavotropic invited Steady
b Irrotational Inviscid bodyforcesareconservative barotropic steady
C Steady Inviscid Constant density Irrotational bodyforces are conservative
d Inviscid barotropic body forces areconservative
3.17 What is the circulation around a circle of constant radius R1 for the velocity eld?
T
given
free
find thecirculation
forces
assumptions inviscid barotropic conservative body
solution
T
ft as
tigerrat
of
p
3.18 The velocity eld for the fully developed viscous ow discussed in Ex. 2.2 is
a Em
y
I
v o
wO
Is the ow rotational or irrotational? Why?
given Velocity field
find if the flow is rotational or Irvotational
assumptions steady incompressible
uniformflow
Solution
If Irrotational then
ft
O
In
ay
My
0
O
rotationalflow
3.21 The absolute value of velocity and the equation of the potential function lines in a two
dimensional velocity eld are given by the expressions
Itt
My
y
twodimensional
velocityfield
q ya ya a
Evaluate both the left-hand side and the right-hand side of the equation to demonstrate the validity of
Stokes’s theorem of this irrotational ow.
given Velocity and potential function
find thevalidity of Stokes'stheorem
assumptions Steady incompressible
T is continuously differentiable
solution
HT di
to T
a
8 25
txt nda
di fo xi.dxi
245
fjzyj.dyjtfizxi
dxitffzyj.dyt to
T di x'to
y't
Halextl vida la
x
i
l
i
y't
i
EE E
vida
4 1 41
0
la loitojl vida
O
3.23 the stream function of a two-dimensional, incompressible ow is given by
Y
E Inr
(a) graph the streamlines
(b) what is the velocity eld represented by this stream function? Does the resultant velocity eld
satisfy the continuity equation?
(c) nd the circulation about a path enclosing the origin. For the path of integration, use a circle of
radius 3 with a center at the origin. How does the circulation depend on the radius?
function
given stream
find
Streamlined Velocity
assumptionJ 2 dimensional
field Circulation
Incompressible
solution
I
b Ur
t O
no Yi Ér
If
rPur
F Epée
ftp.PVotfpvz
0
0 0 0 0 yes satisfies
t
go
i di to
Erra
of
P